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\begin{document}

\title{高等代数一}
\subtitle{09-矩阵的加法、减法、数乘、乘法、幂次、转置}
%\institute{上海立信会计金融学院}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年10月20日} }

\maketitle

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\begin{frame}{内容提要 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{enumerate}
\item  矩阵的概念与记号，矩阵相等的定义
\item  矩阵的加法、减法、数乘、乘法、幂次、转置
\item  一些特殊的矩阵：零矩阵、单位矩阵、数量矩阵、对称矩阵
\item  矩阵乘法的结合律
\item  矩阵乘法与转置的关系
\item  矩阵的行列式、矩阵的迹
\item  把矩阵代入一个多项式
\end{enumerate}

\end{frame}

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\begin{frame}{9.1. 什么是矩阵？}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{itemize}
\item  矩阵是一个正方形或长方形的表格。
\item  一个$m$行$n$列的矩阵称为是 $m\times n$ 阶的。
\item  矩阵的元素通常是实数或复数。
\item  {\color{red} 两个矩阵称为相等，当且仅当它们的阶数相同，对应元素也分别相同。}
\item  矩阵用小括号或中括号表示，例如，一个 $3\times 4$ 阶的矩阵可以写成
{\footnotesize 
\begin{eqnarray*}
A=\begin{pmatrix} *&*&*&* \\ *&*&*&* \\ *&*&*&* \end{pmatrix} 
= \begin{bmatrix} *&*&*&* \\ *&*&*&* \\ *&*&*&* \end{bmatrix}.
\end{eqnarray*}
}
\item  矩阵一般用大写的罗马字母表示，如 $A,B,C$ 等。
\item  矩阵的元素一般用小写的罗马字母表示，如 $a,b,c$ 等。

\end{itemize}

\end{frame}

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\begin{frame}{9.2. 矩阵的加法与数乘}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 两个同阶的矩阵可以相加，结果是分量对应相加。}例如，
{\footnotesize
\begin{eqnarray*}
\begin{pmatrix} a_1&b_1 \\ c_1&d_1 \end{pmatrix} + \begin{pmatrix} a_2&b_2 \\ c_2&d_2 \end{pmatrix} = 
\begin{pmatrix} a_1+a_2&b_1+b_2 \\ c_1+c_2&d_1+d_2 \end{pmatrix}.
\end{eqnarray*}
}
\item  {\color{red} 一个数乘以一个矩阵，等于这个数乘以矩阵的每一个分量。}例如，
{\footnotesize
\begin{eqnarray*}
k\cdot \begin{pmatrix} a&b \\ c&d \end{pmatrix} = \begin{pmatrix} ka&kb \\ kc&kd \end{pmatrix}.
\end{eqnarray*}
}
\item 矩阵的减法是分量对应相减，减法也可以通过数乘与加法组合得到，
{\footnotesize
\begin{eqnarray*}
A - B = A + (-1)\cdot B.
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{9.3. 矩阵的乘法（线性代数方式）}

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\begin{itemize}
\item  {\color{red} 矩阵的乘法可以理解为从``线性方程组''缩写而来，即
{\footnotesize
\begin{eqnarray*}
\left\{\begin{array}{l}
a_1x_1+a_2x_2 = a_3 \\
b_1x_1+b_2x_2 = b_3
\end{array}\right. 
\Longleftrightarrow
\begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} 
=\begin{pmatrix} a_3 \\ b_3 \end{pmatrix}. 
\end{eqnarray*}
}
}

\item  例子1：将下述两个线性方程组写成关于矩阵乘法的一个等式，
{\footnotesize
\begin{eqnarray*}
\left\{\begin{array}{l}
a_1x_1+a_2x_2 = a_3, \\
b_1x_1+b_2x_2 = b_3,
\end{array}\right. 
%\hspace{0.5cm}
\text{  以及  }
\left\{\begin{array}{l}
a_1y_1+a_2y_2 = a_4, \\
b_1y_1+b_2y_2 = b_4.
\end{array}\right. 
\end{eqnarray*}
}
\item  解答：这两个线性方程组的系数矩阵相同，所以可以写成下述形式，
{\footnotesize
\begin{eqnarray*}
\begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix}\begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix} 
=\begin{pmatrix} a_3&a_4 \\ b_3&b_4 \end{pmatrix}. 
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{9.4. 矩阵乘法的第一种简写}

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\begin{itemize}
\item 例子2：若记
{\footnotesize
\begin{eqnarray*}
A= \begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix},
\hspace{0.3cm}
X= \begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix},
\hspace{0.3cm}
B=\begin{pmatrix} a_3&a_4 \\ b_3&b_4 \end{pmatrix},
\end{eqnarray*}
}
\item 则矩阵乘法
{\footnotesize
\begin{eqnarray*}
\begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix}\begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix} 
=\begin{pmatrix} a_3&a_4 \\ b_3&b_4 \end{pmatrix}
\end{eqnarray*}
}
可以简写成
\begin{eqnarray*}
AX=B.
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{9.5. 矩阵乘法的第二种简写}

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\begin{itemize}
\item  例子3：若记
{\footnotesize
\begin{eqnarray*}
\xi = \begin{pmatrix} x_1 \\ x_2  \end{pmatrix},
\hspace{0.3cm}
\eta = \begin{pmatrix} y_1 \\  y_2 \end{pmatrix},
\end{eqnarray*}
}
\item 则矩阵乘法
{\footnotesize
\begin{eqnarray*}
AX=\begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix}\begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix} 
=\begin{pmatrix} a_3&a_4 \\ b_3&b_4 \end{pmatrix} = B
\end{eqnarray*}
}
可以简写成
{\footnotesize
\begin{eqnarray*}
AX = A\begin{pmatrix} \xi & \eta \end{pmatrix} = \begin{pmatrix} A \xi & A \eta \end{pmatrix} = B.
\end{eqnarray*}
}
\item  {\color{red} 这相当于把矩阵 $X$ 按左、右分块，把结果 $B$ 也按左、右分块。} 

\end{itemize}

\end{frame}

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\begin{frame}{9.6. 矩阵乘法的第三种简写}

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\begin{itemize}
\item  例子4：若记
{\footnotesize
\begin{eqnarray*}
\alpha= \begin{pmatrix} a_1 & a_2 \end{pmatrix},
\hspace{0.3cm}
\beta= \begin{pmatrix} b_1 & b_2 \end{pmatrix},
\end{eqnarray*}
}
\item 则矩阵乘法
{\footnotesize
\begin{eqnarray*}
AX=\begin{pmatrix} a_1&a_2 \\ b_1&b_2 \end{pmatrix}\begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix} 
=\begin{pmatrix} a_3&a_4 \\ b_3&b_4 \end{pmatrix} = B
\end{eqnarray*}
}
可以简写成
{\footnotesize
\begin{eqnarray*}
AX = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} X = \begin{pmatrix} \alpha X \\  \beta X \end{pmatrix} = B.
\end{eqnarray*}
}
\item  {\color{red} 这相当于把矩阵 $A$ 按上、下分块，把结果 $B$ 也按上、下分块。}

\end{itemize}

\end{frame}

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\begin{frame}{9.7. 某些矩阵的另一个名称}

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\begin{itemize}
\item  {\color{red} 一个 $1\times n$ 阶的矩阵也称为一个 $n$ 维的行向量。}例如，
{\footnotesize
\begin{eqnarray*}
\alpha = \begin{pmatrix} a_1 & a_2 & \cdots & a_n \end{pmatrix}.
\end{eqnarray*}
}
\item  {\color{red} 一个 $n\times 1$ 阶的矩阵也称为一个 $n$ 维的列向量。}例如，
{\footnotesize
\begin{eqnarray*}
\beta = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}.
\end{eqnarray*}
}
\item 一般用小写的希腊字母来表示行向量和列向量，如 $\alpha, \beta, \gamma$ 等。

\end{itemize}

\end{frame}

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\begin{frame}{9.8. 矩阵的乘法没有交换律}

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\begin{itemize}
\item  {\color{red}两个矩阵相乘，前后次序很重要，因为 $AB$ 一般不等于 $BA$. } 

\item  例子5：找出两个矩阵，使得 $AB\neq BA$. 

\item  解答：这样的例子很多。例如，
{\footnotesize
\begin{eqnarray*}
AB &=& \begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix}
\begin{pmatrix} 5&6 \\ 7&8 \end{pmatrix}=
\begin{pmatrix} 19&22 \\ 43&50 \end{pmatrix}, \\ 
BA &=& \begin{pmatrix} 5&6 \\ 7&8 \end{pmatrix}
\begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix}=
\begin{pmatrix} 23&34 \\ 31&46 \end{pmatrix}.
\end{eqnarray*}
}
\item  {\color{red}矩阵的乘法没有交换律，这个很重要。}


\end{itemize}

\end{frame}

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\begin{frame}{9.9. 一般阶数的矩阵乘法}

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\begin{itemize}

\item 有时候，$AB$ 可以相乘，但是 $BA$ 不一定可以相乘。

\item  {\color{red}当 $A$ 是一个 $m\times n$ 阶的矩阵，$B$ 是一个 $n\times k$ 阶的矩阵的时候，矩阵的乘积 $AB$ 有意义，且为一个 $m\times k$ 阶的矩阵。}

\item  例子6：计算下述矩阵的乘积，
{\footnotesize
\begin{eqnarray*}
AB=\begin{pmatrix} 1&1&1 \\ 2&2&2 \end{pmatrix}
\begin{pmatrix} 1&2&3&4 \\ 1&2&3&4 \\ 1&2&3&4  \end{pmatrix}=
\begin{pmatrix} 3&6&9&12 \\ 6&12&18&24 \end{pmatrix}=C.
\end{eqnarray*}
}
\item  {\color{red} 一般地，设 $A=(a_{ij})_{m\times n}$, $B=(b_{ij})_{n\times k}$, 则 $AB=C=(c_{ij})_{m\times k}$, 其中 
\begin{eqnarray*}
c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj}.
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{9.10. 矩阵的乘法有结合律}

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%每页详细内容

\begin{itemize}
\item  {\color{red} 定理：矩阵的乘法有结合律，即
\begin{eqnarray*}
(AB)C = A(BC)
\end{eqnarray*}
对所有可以相乘的矩阵 $A,B,C$ 都成立。
}

\item 证明：按矩阵乘法的定义，验证两边矩阵的元素都对应相等。
    \begin{enumerate}
    \item 首先看到等式两边的矩阵的阶数一致。
    \item 计算左边矩阵 $(AB)C$ 的每个分量。
    \item 计算右边矩阵 $A(BC)$ 的每个分量。
    \end{enumerate}

\end{itemize}

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\begin{frame}{9.11. 矩阵的转置 }

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%每页详细内容

\begin{itemize}
\item  {\color{red}一个 $m\times n$ 阶的矩阵 $A$, 转置得到一个 $n\times m$ 阶的矩阵 $A^t$. } 
\item  {\color{red}矩阵 $A^t$ 的第 $(i,j)$ 元素（即第$i$行、第$j$列）是矩阵 $A$ 的第 $(j,i)$ 元素。} 
\item 矩阵 $A$ 的转置记为 $A^t$ 或 $A^T$ 或 $A^\prime$. 例如，
{\footnotesize 
\begin{eqnarray*}
A=\begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \end{pmatrix},
\hspace{0.3cm}
A^t=\begin{pmatrix} a_1&b_1 \\ a_2&b_2 \\  a_3&b_3 \end{pmatrix}. 
\end{eqnarray*}
}
\item  {\color{red}定理：矩阵的转置与乘法的关系为 $(AB)^t = B^tA^t$. }

\item 证明思路：验证等式两边的矩阵，元素对应相等。


\end{itemize}

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\begin{frame}{9.12. 对称矩阵 }

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\begin{itemize}

\item  {\color{red}如果 $A^t=A$, 那么称 $A$ 是对称矩阵。 }

\item  例子7：写出2阶、3阶和4阶对称矩阵的一般形式。

\item  解答：
{\footnotesize 
\begin{eqnarray*}
A=\begin{pmatrix} a&b \\ b&c \end{pmatrix},
\hspace{0.3cm}
A=\begin{pmatrix} a&b&c \\ b&d&e \\  c&e&f \end{pmatrix},  
\hspace{0.3cm}
A=\begin{pmatrix} x&a&b&c \\ a&y&d&e \\  b&d&z&f \\ c&e&f&w \end{pmatrix}. 
\end{eqnarray*}
}

\item  {\color{red} 矩阵 $A=(a_{ij})_{m\times n}$ 是对称矩阵，当且仅当 $m=n$ 且 $a_{ij}=a_{ji}$ 对所有 $1\le i\le n$ 与 $1\le j\le n$ 都成立。 }


\end{itemize}

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\begin{frame}{9.13. 一些特殊的矩阵：零矩阵}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}元素都是零的矩阵称为零矩阵，记为 $O_{m\times n}$, 或者简记为 $O$. }
\item  不同阶数的零矩阵是不相同的，例如，
{\footnotesize
\begin{eqnarray*}
&& O_{1\times 1}=0, 
\hspace{0.3cm}
O_{1\times 2}=\begin{pmatrix} 0&0  \end{pmatrix}, 
\hspace{0.3cm}
O_{2\times 1}=\begin{pmatrix} 0 \\ 0 \end{pmatrix}, 
\hspace{0.3cm}
O_{2\times 2}=\begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix}, \\
%\hspace{0.3cm}
&& O_{2\times 3}=\begin{pmatrix} 0&0&0 \\ 0&0&0 \end{pmatrix},
O_{3\times 2}=\begin{pmatrix} 0&0 \\ 0&0 \\ 0&0\end{pmatrix}.
\end{eqnarray*}
}
\item  零矩阵与其它矩阵如果可以相乘，那么结果总是零矩阵。例如，
{\footnotesize
\begin{eqnarray*}
O_{2\times 2}A = \begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix}\begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \end{pmatrix}
=\begin{pmatrix} 0&0&0 \\ 0&0&0 \end{pmatrix}=O_{2\times 3}.
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{9.14. 一些特殊的矩阵：单位矩阵}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}对角线元素是1，其余元素是0的方形矩阵，称为单位矩阵。}
\item  {\color{red}单位矩阵记为 $E_n$ 或 $I_n$ 或 $E$ 或 $I$. } 例如，
{\footnotesize
\begin{eqnarray*}
E_{1}=1, 
\hspace{0.3cm}
E_{2}=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}, 
\hspace{0.3cm}
E_{3}=\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{pmatrix}.
\end{eqnarray*}
}
\item 单位矩阵与其它矩阵如果可以相乘，那么结果总是那个矩阵。例如，
{\footnotesize
\begin{eqnarray*}
E_{2}A = \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}\begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \end{pmatrix}
=\begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \end{pmatrix}=A.
\end{eqnarray*}
}
\item  {\color{red}数量矩阵是对角线元素都相同，其余元素是0的方形矩阵。}例如，
{\footnotesize
\begin{eqnarray*}
kE_{2}=\begin{pmatrix} k&0 \\ 0&k \end{pmatrix}.
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{9.15. 方形矩阵的行列式和迹}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}一个方形矩阵 $A$ 的行列式(determinant)记为 $|A|$ 或 $\det(A)$. }
\item  {\color{red}一个方形矩阵 $A$ 的迹(trace)是指它的对角线的元素的和，记为 $\text{tr}(A)$. }
\item  例子8：设 $A$ 是下述矩阵，求它的行列式与迹，
{\footnotesize
\begin{eqnarray*}
A=\begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3   \end{pmatrix}. 
\end{eqnarray*}
}
\item  解答：它的行列式和迹分别为
{\footnotesize
\begin{eqnarray*}
\det(A)=\begin{vmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3   \end{vmatrix}, 
\hspace{0.3cm}
\text{tr}(A)=a_1+b_2+c_3. 
\end{eqnarray*}
}

\end{itemize}

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\begin{frame}{9.16. 把矩阵代入一个多项式}

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%每页详细内容

\begin{itemize}
\item 例子9：设矩阵 $A$ 和多项式 $f(x)$ 如下，求 $f(A)$.
{\footnotesize
\begin{eqnarray*}
A=\begin{pmatrix} 1&2 \\ 3&4  \end{pmatrix}, 
\hspace{0.3cm}
f(x) = x^2+2x+3.
\end{eqnarray*}
}
\item 解答：注意常数项要乘以单位阵，
{\footnotesize
\begin{eqnarray*}
 f(A) &=& A^2+2A+3E_2 \\
 &=&\begin{pmatrix} 1&2 \\ 3&4  \end{pmatrix}\begin{pmatrix} 1&2 \\ 3&4  \end{pmatrix} 
 + 2\begin{pmatrix} 1&2 \\ 3&4  \end{pmatrix} + 3\begin{pmatrix} 1&0 \\ 0&1  \end{pmatrix} \\ 
&=& \begin{pmatrix} 7&10 \\ 15&22  \end{pmatrix} + \begin{pmatrix} 2&4 \\ 6&8  \end{pmatrix} 
+ \begin{pmatrix} 3&0 \\ 0&3  \end{pmatrix} \\
 &=& \begin{pmatrix} 12&14 \\ 21&33  \end{pmatrix}.
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\item  设有三个2阶矩阵
{\footnotesize
\begin{eqnarray*}
A= \begin{pmatrix} a_1&a_2 \\ a_3&a_4 \end{pmatrix},
\hspace{0.3cm}
B= \begin{pmatrix} b_1&b_2 \\ b_3&b_4 \end{pmatrix},
\hspace{0.3cm}
C=\begin{pmatrix} c_1&c_2 \\ c_3&c_4 \end{pmatrix}. 
\end{eqnarray*}
}
\begin{enumerate}
\item[1.1.]  验证矩阵乘法的结合律 $(AB)C=A(BC)$. 
\item[1.2.]  验证矩阵乘法与转置的关系 $(AB)^t = B^tA^t$.
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\item  证明思路：分别计算等式的两边。验证2阶矩阵的4个分量对应相等。


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